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A dynamic efficiency wage model with learning by doing
a b ,
*
Mohamed Jellal , Yves Zenou
a
´
Universite de Rabat, Rabat, Morocco b
´ ´
CERAS, Ecole Nationale des Ponts et Chaussees and GAINS, Universite du Maine, Paris, France
Received 5 March 1999; accepted 26 July 1999
Abstract
We propose a dynamic efficiency wage model with learning by doing. By taking into account the change in the stock of workers’ knowledge, firms set efficiency wages such that the effort–wage elasticity is not in general equal to one. 2000 Elsevier Science S.A. All rights reserved.
Keywords: Solow condition; Effort function; Learning by doing
JEL classification: J41
1. Introduction
The efficiency wage literature (Akerlof and Yellen, 1986) stipulates that there is a direct and increasing relation between the wage paid by firms and the effort provided by workers: the greater the wage, the larger the effort. The model is in general static and maximizing profit firms set an efficiency wage such that the effort–wage elasticity is equal to unity. This is referred to as the Solow condition (see Solow, 1979).
It has been argued, however, that this condition does not hold in general. In their introduction, Akerlof and Yellen (1986) stipulate that the effort–wage elasticity is less than one. Different reasons have been given in the literature. Schmidt-Sørensen (1990) has introduced fixed employment costs per worker (such as for example employer-provided health insurance) in the profit function; Pisauro (1991) has taken into account specific taxes on labor; Rasmaswamy and Rowthorn (1991) have dropped the assumption of the labor augmenting production function and have used a general one; Lin and Lai (1994) have proposed a dynamic model with (external) turnover costs; and Jellal and Zenou (1999) have focused on the role of uncertain job matching on workers’ effort. They have all shown that the Solow condition does not hold and in general the effort–wage elasticity is less than one.
*Corresponding author. Tel.: 133-1-4458-2870; fax:133-1-4458-2880.
E-mail address: zenou@worldnet.fr (Y. Zenou)
In this paper, we develop a dynamic model of efficiency wage where firms take into account both static and dynamic gains of high wages. We assume that there is some learning by doing so that workers accumulate a stock of knowledge which allows them to increase their effort. This is in the same spirit as in Arrow (1962), since learning by doing is here the steady improvement engendered by the experience of producing.
We show that our dynamic efficiency wage can be greater, lower or equal to the static one (for which the elasticity of effort with respect to wage is unitary) depending on a trade off between productivity and static and dynamic gains. We believe that the work relation is a long run one so that firms must induce workers’ effort at each period of time.
The remainder of the paper is as follows. The next section describes the basic model. In Section 3, we develop the dynamic analysis and focus on the steady-state solution. Section 4 concludes.
2. The basic model and notations
Consider an infinitely lived representative firm and N homogeneous workers. As it is standard in the efficiency wage literature, the firm takes into account the role of the wage offer as an incentive to increase labor efficiency. Let us denote by e(w) the effort function and w the wage offer, with e9(w)
.0 and e0(w),0. In this context, the production function can be written as:
Yt5F X(A ,e(w )),N
s
t t td
(1)where Y is the output at time t, N is the number of workers hired at time t, X(.) corresponds to thet t efficiency function per worker which depends on the knowledge (specific or not) A accumulated att
1
time t and on the flow of effort e(w ). We assume that the production function is such that:t
9
9
9
9
so that F(.) is both increasing and concave in w and N. It is important to observe that this production function is quite different than the one used in general in the efficiency wage model (see Solow, 1979) in which: Y5F e(w)N . Indeed, in the standard static approach, wages have a direct and instantaneouss d
effect on effort, and thus on the output. Here, we want to emphasize the fact that effort can vary from one period to another so that the control variable w used by the firm must induce workers at eacht
1 9
In terms of notation, FX5 ≠F /≠X. We use the same notation for all partial derivatives.
period of time. We also want to put forward the fact that there is some learning by doing in the efficiency function and thus in the output: the more knowledge is accumulated, the greater is the efficiency and the output. Indeed, knowledge is growing in time, and learning is the product of experience. In order to be more precise, we assume that productivity (or knowledge) per worker is given by the following evolution equation:
~
At5G e(w )
s
td
2dAt (2)where d.0 is the rate of depreciation of the stock of knowledge. We assume that:
9
99
Ge.0, Ge ,0
3
so that G(.) is increasing and concave in w. Eq. (2) means that there are two aspects that affect the evolution of the stock of knowledge. On one hand, the wage policy at each period induces workers to be more efficient and thus to learn more. On the other, and independently of the wage policy, the stock of knowledge depreciates over time because workers forget a part of what they have learned. Thus, Eq. (2) implies that change in the stock of knowledge of the worker in the immediate future depends partially on his effort and on the rate of depreciation of the stock of knowledge.
3. The steady-state equilibrium
In this context, the wage rate has an impact on current effort and thus on productivity through the function F(.) as well as on the intertemporal efficiency through G(.). Therefore, a firm would like to set a wage policy that incorporates both static and dynamic gains. For that, the firm solves the following problem:
where r is the subjective discount rate. We have typically here an optimal control problem where the control variables are w and N and the state variable is A . If we denote byt t t lt the costate variable associated with A , the Hamiltonian function H can be expressed as:t
The first two terms of the RHS of (4) correspond to the immediate profit from the choice of w andt N while the last term is the value of the ‘investments’ that are affected by w and N . According to thet t t
maximum principle of optimal control theory, the control variables w and N must be chosen so as tot t
maximize H at each date, given the current values of the state and costate variables, A andt lt. If one assumes continuous differentiability of the Hamiltonian, this implies that:
≠H ≠F ≠X
Furthermore, the optimal control theory (see, e.g., Kamien and Schwartz, 1981) imposes two necessary conditions (that are also sufficient since H is concave in A , w and N ) that the costatet t t variable l must satisfy. The first one is the Euler equation which states that:
≠F ≠X
~ ] ]]
lt5(r1d)lt2≠X ≠A (7)
t
and the second one consists of the following transversality condition:
2rt
lim e ltAt50 (8)
t→1`
In this paper, we are interested in the steady-state solution of this problem. The stationary state to this growth model is when both A (the state variable) and the shadow valuet lt (costate variable) are
~ ~
constant, i.e., A50 and l50. It is important to observe that there is a unique (steady-state) solution
~ ~
to this dynamic system which is the saddle point (A*,l*) solution of the system A50 and l50. Indeed, from (5) and (6), we get w and N as a function of A and l. By inserting these two values in (2) and (7), we obtain a two-dimensional system of differential equation in A and l. By using the initial condition A(0)5A0$0 and the transversality condition (8) which determines a terminal condition, there are just enough boundary conditions to determine a unique solution to the dynamic
4
system; it will be the one that converges asymptotically to the stationary state (A*,l*). At the steady-state solution, we have (drop the time index):
~
Thus, by using (10), we can rewrite (5) as:
≠F ≠X 1 ≠F ≠X
] ]≠X ≠ee9(w)1
S D
]] ] ]r1d ≠X ≠AG9(.)e9(w)5N (11)4
It can further be shown that this steady-state equilibrium (A*,l*) is locally stable. Indeed, by linearizing the two
~ ~
Eq. (11) has the following intuitive interpretation. The first term of the LHS is the static marginal gain of raising wages and the second term is the intertemporal marginal gain. The RHS of (11), N, is of course the marginal cost of a wage increase. Thus, when the firm has to decide its wage policy, it faces a trade off between its marginal gain (both today and tomorrow) and its marginal cost. The interesting feature here is that the firm must take into account that the flow of effort varies over time. Now, by using (6) and (9), and after some manipulations, (11) becomes:
e(w) e9(w) e(w)G9(e) e9(w)
wherehA andhe are the elasticity of efficiency with respect to knowledge and to effort, respectively;
hX and hN represent the elasticity of the production with respect to efficiency and to employment, respectively;uGis the elasticity of knowledge accumulation with respect to effort and finally,´w is the (standard) elasticity of effort with respect to wage. By using these definitions, we can rewrite (12) as:
d
]]
´ hw
S
e.hX1 hA?hX?uGD
5hN (13)d 1r
Proposition 1. The long run optimal efficiency wage w* is given by:
h
We are now in position to discuss whether the Solow condition (i.e., ´w51) is valid.
Proposition 2. From Proposition 1, we have the following equivalence:
hN d the ratio of the elasticity of productivity of output while the RHS is the global gain of incentive (he is the static gain emanating from the flow of effort and
d
]]
uG
S
d1rD
hA,then firms have no interest to set high wages (´w.1), whereas wages are naturally high (´w,1) if we have the reverse inequality.
Another interesting implication of Proposition 2 is that the dynamic efficiency wage (14) can be
greater,equal or lower than the static one (in which ´w51). This result contradicts that of Barnerji and Gupta (1997) where the dynamic efficiency wage is always greater than the static one or stated differently where´w,1 (see their Proposition 1). This is due to the fact that they focus on a different issue, namely the problem of health accumulation in LDCs. In their paper, the efficiency function is
h(t)5h[w(t),e(t),a1], where a1 is a parameter and effort e(.) is identified with health, and the
~
evolution equation is e5g[w(t),a2]2de where a2 is another parameter. Therefore, in Barnerji and Gupta (1997) a dynamic high wage is needed since health accumulation depends drastically on it.
We have, therefore, obtained a general result that formalizes in a dynamic context the statement of Stiglitz (1987). He indeed argues that many of the results in the efficiency wage theory depend crucially on the existence of some region(s) where an increase in the wage leads to more than proportionate increases in the work effort. We show that there are also some regions where a wage raise generates an effort lower or equal to the initial wage increase (a wage for which the elasticity of effort with respect to wage is greater than 1 is obviously lower than a wage characterized by an elasticity less or equal to 1).
It is now interesting to discuss the following special cases. IfhA50, which means that production requires simple tasks of routine without requiring qualification and accumulation of knowledge, and
he5 hN5 hX51, we are back to the Solow case with ´w51. If hA50 andhX51, we obtain exactly the result of Rasmaswamy and Rowthorn (1991) in which:
hN
]
´w_1⇔h _1
e
This shows that our analysis is quite general so that it can encompass other papers as special cases.
4. Conclusion
The aim of this paper is to highlight the fact that the work relation is a long run one so that firms must induce workers’ effort at each period of time. In a dynamic framework with a learning by doing process, we generalize the Solow condition (static efficiency wage) in which the wage elasticity is equal to one and show that our dynamic efficiency wage can be greater, equal or lower than the static one.
References
Akerlof, G.A., Yellen, J.L., 1986. In: Efficiency Wage Models of the Labor Market, Cambridge University Press, Cambridge. Arrow, K.J., 1962. The economic implications of learning-by-doing. Review of Economic Studies 29, 155–173.
Barnerji, S., Gupta, M.R., 1997. The efficiency wage given long-run employment and concave labor constraint. Journal of Development Economics 53, 185–195.
Kamien, M.I., Schwartz, N.L., 1981. In: Dynamic Optimization. The Calculus of Variations and Optimal Control in Economics and Management, North-Holland, Amsterdam.
Lin, C.-C., Lai, C.-C., 1994. The turnover costs and the Solow condition in an efficiency wage model with intertemporal optimization. Economics Letters 45, 501–505.
Pisauro, G., 1991. The effects of taxes on labour in efficiency wage models. Journal of Public Economics 46, 329–345. Rasmaswamy, R., Rowthorn, R.E., 1991. Efficiency wages and wage dispersion. Economica 58, 501–514.
Schmidt-Sørensen, J.B., 1990. The equilibrium effort-wage elasticity in efficiency wage models. Economics Letters 32, 365–369.
Solow, R.M., 1979. Another possible source of wage stickiness. Journal of Macroeconomics 1, 79–82.
